A short post, intended to convey the philosophy behind Math in a Month(TM)’s approach to homework problems.
Two ways in which the title applies:
- If you don’t solve problems that are somewhat difficult, you don’t really “sharpen the blade” of your mind. You’re instead getting credit for filling out forms. We don’t do that here. Learning math by rote might get someone a college degree, and a job, but won’t contribute anything of any value to technology or society through the sacred time we have together to learn math. So math is a form of training your mind to sharpen the blade.
- Math itself is not allowed, under the current schooling model, to meaningfully reform because we cannot pit math problems against one another.
You probably guessed what #1 was saying before you finished reading it. Self explanatory.
But #2 is more intriguing to me. It led me to the following question:
What if you only could teach a year’s worth of math in 100 problems?
Think about it: we have to pick the 100 top problems worth studying.
Student time is limited: not only are there distractions, but more important life lessons need to be learned as well. We can’t waste the entire school year learning math by rote, and then call it a day and say “up to you now, kid.”
Instead, we need to extract maximum pedagogical value per problem. Yes, that’s nerdy-sounding, but it’s what we must do.
Recently deceased Princeton (& Cambridge) Mathematician John Conway once said that mathematics to him was a collection of interesting examples, organized by the theorems that help categorize the examples, and bring insight to bear on the problems faced. Often, the motivation for why the math results as it does makes sense from that angle: so choosing problems wisely to give students helps them see and extract that value.
In short, we have students do no more than 5 homework problems. Problems you expect someone to remember afterward. When I write the curriculum, I make the problems
- distinctive – humorously presented, or showcasing an intriguing situation
- iterative – meaning after solving it, you can envision how you would try to solve a problem of bigger size or complexity, using the same basic rules
- foundational – you will be able to solve many other problems after learning strategies for this one (not true of most math HW problems, nearly all are extraneous to the student’s life and career)
- pedagogically robust – the examples help teach the tricks involved, and things to watch out for. Ideal training examples, unlike most math textbooks, which certainly help with automaticity (if you do them all, as I did starting in my Calc BC class)
- interesting to teachers/adults – even those with lots of experience will appreciate the perspectives Math in a Month(TM) brings to the table.
Math must vet its homework. We do that for you, and offer you nothing but the most interesting, most evocative, most pedagogical math problems in all our work and curricula.